The numerical simulations of the dynamics of high Reynolds number (\(Re>100{,}000\)) swirling flows in pipes with varying geometries of engineering applications continues to be a challenging computational problem, specifically when vortex-breakdown zones or wall-separation regions naturally evolve in the flows. To tackle this challenge, the present paper describes a simulation scheme of the evolution of inviscid, axisymmetric and incompressible swirling flows in expanding or contracting pipes. The integration in time of the circulation together with azimuthal vorticity uses an explicit, first-order accurate finite-difference scheme with a second-order accurate upwind difference formulation in the axial and radial directions. The Poisson solver for advancing in time the spatial distribution of the stream function as a function of azimuthal vorticity uses a second-order accurate over-relaxation difference scheme. No additional numerical steps are needed for computing the natural evolution of flows including the dynamics to states with slow-speed recirculation zones along the pipe centerline or attached to pipe wall. This numerical method shows convergence of the computed results with mesh refinement for various swirl levels and pipe geometry variations. The computed results of time-asymptotic states also present agreement with available theoretical predictions of steady vortex flows in diverging or contracting pipes. In addition, comparison with available experimental data demonstrates that the present algorithm accurately predicts instability processes and long-term mean-flow dynamics of vortex flows in pipes at high Re. The inviscid-flow simulation results support the theoretical predictions and clarify the nature of high-Re flow evolution.

中文翻译：

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具有各种几何形状的圆形管道中的轴对称无粘性涡流模拟

具有不同工程应用几何形状的管道中高雷诺数 (\(Re>100{,}000\)) 涡流动力学的数值模拟仍然是一个具有挑战性的计算问题，特别是当涡破区域或壁面分离区域在流动中自然演化。为了应对这一挑战，本文描述了膨胀或收缩管道中无粘性、轴对称和不可压缩涡流演化的模拟方案。环流时间与方位涡度的积分使用明确的、一阶精确的有限差分格式，在轴向和径向上使用二阶精确的逆风差分公式。用于将流函数的空间分布作为方位涡度的函数及时推进的泊松求解器使用二阶精确过松弛差分方案。无需额外的数值步骤来计算流动的自然演变，包括沿管道中心线或连接到管壁的低速再循环区状态的动力学。这种数值方法显示了计算结果与各种涡流水平和管道几何变化的网格细化的收敛性。时间渐近状态的计算结果也与现有的对扩张或收缩管道中稳定涡流的理论预测一致。此外，与现有实验数据的比较表明，本算法准确地预测了高 Re 下管道中涡流的不稳定过程和长期平均流动力学。无粘流模拟结果支持理论预测并阐明了高 Re 流演化的性质。